Everettian relative states in the Heisenberg picture (2008.02328v3)

Abstract: Everett's relative-state construction in quantum theory has never been satisfactorily expressed in the Heisenberg picture. What one might have expected to be a straightforward process was impeded by conceptual and technical problems that we solve here. The result is a construction which, unlike Everett's one in the Schr\"odinger picture, makes manifest the locality of Everettian multiplicity, and its inherently approximative nature, and its origin in certain kinds of entanglement and locally inaccessible information. Our construction also allows us to give a more precise definition of an Everett 'universe', under which it is fully quantum, not quasi-classical, and we compare the Everettian decomposition of a quantum state with the foliation of a spacetime.

• The paper establishes a robust framework for formulating Everettian relative states in the Heisenberg picture, emphasizing local observable dynamics over non-local state evolution.
• The authors employ quantum computational networks and precise mathematical formalisms to demonstrate how qubits can encode local quantum information.
• The study provides key insights into quantum measurements and decoherence, paving the way for future applications in quantum gravity and advanced quantum theories.

Everettian Relative States in the Heisenberg Picture: A Comprehensive Summary

The paper "Everettian Relative States in the Heisenberg Picture" by Samuel Kuypers and David Deutsch seeks to rectify the longstanding difficulty of formulating Everett's relative-state construction within the Heisenberg picture of quantum mechanics. This issue is of significant importance, as expressing Everett's construction exclusively in the Schrödinger picture could potentially undermine the entirety of Everettian quantum theory, particularly if future theoretical frameworks, such as quantum gravity, were to rely solely on Heisenberg formalism.

Core Objective

The main objective of the paper is to establish a robust framework for Everett's relative-state construction within the Heisenberg picture. Unlike the Schrödinger picture, where the non-local nature of Everettian "parallel universes" is apparent, the Heisenberg approach emphasizes the intrinsic locality and the approximate nature of these universes. The authors' formulation provides a more precise conceptualization of an Everett "universe," demonstrating its status as fully quantum rather than quasi-classical.

Key Results and Methodology

The paper delineates the differences between the Heisenberg and Schrödinger pictures, particularly how quantum observables and the state vector evolve. In the Heisenberg picture, observables are treated as time-evolving entities, while the state vector remains fixed, highlighting local interactions without invoking the non-local attributes of the Schrödinger state vector. The authors construct Heisenberg relative states for a precisely defined quantum computational network, arguing that the properties they derive are generalizable across all quantum systems because of the universality of quantum computational networks.

Key mathematical formalisms are introduced to describe how qubits can encode information through observable descriptors that are local in nature. By utilizing a network of qubits, the authors illustrate how relative states can be established, manipulated through quantum gates (e.g., the not gate, cnot gate), and how such states evolve over time. The Heisenberg picture elucidates information flow and provides a comprehensive view of the entanglement's role in quantum mechanics, without the need for non-local interactions suggested by the Schrödinger approach.

Implications

The implications of expressing Everettian relative states in the Heisenberg picture stretch across theoretical and practical domains in quantum theory:

• Locality and Quantum Measurement: By localizing Everettian universes, the research aligns with decoherence theories and emphasizes the significance of locality in quantum measurements.
• Quantum Information: The transparency of information location in entangled systems, enabled by the Heisenberg picture, offers new routes for exploring quantum information theory.
• Quantum Gravity and Beyond: This framework may hold value in addressing the challenges of formulating coherent models of quantum gravity or other potential successor theories that operate outside the Schrödinger picture.

Future Directions

The Heisenberg picture rendering of Everettian quantum mechanics opens pathways for future inquiry that can explore:

• Complex Quantum Systems: Extending the model to encompass more complex quantum systems than the basic qubit structures utilized in the paper.
• Cosmological Applications: Evaluating the implications of bounded Everett universes in cosmology and quantum field theory.
• Experimental Validation: Developing experiments that attempt to confirm the conclusions drawn from the Heisenberg-based formulation of Everettian quantum mechanics.

In conclusion, this paper effectively bridges a critical gap in the conceptual structure of Everettian quantum mechanics, offering a locally-grounded, mathematically rigorous alternative to the prevailing Schrödinger-dependent approaches. Thereby, it enhances our understanding of the complexities involved in expressing universally consistent quantum theories and sets a foundation for potential advancements in quantum mechanics.